I don't know if this question is more appropriate for the academia stack exchange, but I'm posting it here because it's more closely related to math itself.

I'm not actually an undergraduate, I'm a 10th grader. I've made a few posts about 3 weeks ago about finding research opportunities in math right now, but then I kind of forgot about them because of a reason I'll mention very soon. I live near a large university that I'll attend next year via dual-enrollment, so I think my best chance is to cold mail one of the math professors there for an opportunity (I cannot attend a math camp like SUMaC for a variety of reasons mentioned in the other posts).

So why haven't I mailed them? Well, I don't even know what I want to research, and I think that it would be a good idea to have a topic in mind in the email. In fact, I don't even know what topics there are. I'm currently working through Introduction to Abstract Algebra by Nicholson and I think I may be able to finish it in a few months (I'm currently learning about the Sylow theorems). I doubt this is even enough background to work on anything in contemporary research, but I'm not entirely sure.

Should I just be patient and learn more before approaching a professor? If not, can anyone give me advice on how to start my journey of selecting a topic?